Definitely sounds like a power problem. I had similar behaviour when I was using an iffy power supply with RasPi. For example, when I started up gcc the screen would get little green dots and sometimes blank out, but would come back when the compile was done.

You may be able to fix the problem following the Troubleshooting Power Supplies advice drgeoff links to. The usual culprits are iffy Micro USB cables and sometimes polyfuse F3. However, if you're powering your HDMI to VGA converter from RasPi it could be that you're getting too much drop across a working polyfuse F3 when you add the current demands of RasPi plus the converter.

Some people have hacked their HDMI to VGA converters to use external power so that it doesn't go through RasPi.

If you have a model 2.0 RasPi (or 1.0 with 0 Ohm jumpers in place of polyfuses F1 and F2) you might try back-powering RasPi from a powered hub instead of using the Micro USB plug. This avoids the voltage drop across F3, though it does lose the protection offered by F3.